1. Field of the Invention
The invention is generally related to a method using a seminal Multi-point statistics (MPS) algorithm to generate numerical pseudocores from digital rock or core samples and borehole-imaging logs. More particularly, this patent specification relates to creating 3D numerical cores from computed X-ray tomography (CT scans) and formation micro-image (FMI) logs, and performing flow modeling in these numerical cores to understand fluid-flow paths and recovery factors in selected reservoir.
2. Background of the Invention
Electrical and acoustic borehole-imaging tools are widely used to log subsurface boreholes to locate and map the boundaries between rock layers, e.g., bed boundaries, and to visualize and orient fractures and faults. Because electrical logging tools are pad-type devices with fixed arrays of electrodes, it is common to have gaps with missing information between the pads. Electrical and acoustic logs commonly have intervals with poor data quality due to non-functioning electrodes, insufficient pad pressure, borehole irregularities, rock debris, decentralized tools, or poor acoustic reflections.
Digital rock models are constructed from 2D thin sections, scanning-electron-microscope (SEM) images, computer-generated sphere packs, laser-scanning confocalmicroscope-images, and various types of CTscans, e.g., conventional, microCT, and synchrotron-computed microtomography. CTscans are the most widely used approach. CTscans are 2-dimensional (2D) cross sections generated by an X-ray source that rotates around the sample. Density is computed from X-ray attenuation coefficients. Scans of serial cross sections are used to construct 3D images of the sample. Because the density contrast is high between rock and fluid-filled pores, CT images can be used to visualize the rock-pore system. Resolutions are on the sub-millimeter to micron scale, depending on the device being used.
Multi-point statistics (MPS) are used to create simulations of spatial geological and reservoir property fields for subsurface reservoir modeling. These methods are conditional simulations that use known results, such as those measured in wellbores, as fixed or “hard” data that are absolutely honored during the simulations. MPS uses 1D, 2D, or 3D “training images” as quantitative templates to model subsurface property fields. MPS modeling captures geological structures from training images and anchors them to data locations. These structures can be either a priori geological interpretations or conceptual models.
Multipoint geostatistics (MPS) is a new advanced geostatistics approach. It allows reservoir modelers to incorporate their prior knowledge, interpretations, or conceptual models into the reservoir modeling process through training images. These training images are numerical representations of the structures/features that are believed to exist in the reservoir under study. Once we have the training images, MPS can extract curvilinear structures or complex features from the training images and anchor them to the reservoir locations where the samples/observations are collected, leading to more realistic reservoir models. Introducing training images into reservoir modeling is a milestone. Note that there are two ingredients in the use of MPS: training images (conceptual models) and the real data (observations). These two pieces are typically separated. However, in realistic applications, generating representative training images, in particular in 3D, has proved to be a bottleneck in MPS applications. Generating a continuous variable training image is even more difficult than the creation of categorical training image.
There are different types of electrical and acoustic borehole-imaging tools used to log subsurface boreholes to locate and map the boundaries between rock layers, e.g., bed boundaries, and to visualize and orient fractures and faults.
For example, electrical borehole images may run in water-based (conductive) mud, such as Schlumberger's FMI (Formation MicroImager) log, which is based on dipmeter technology that has been commercially available since the 1950's. Electrical borehole-imaging tools are, in essence, sophisticated dipmeters. The imaging tools have microresistivity electrodes arranged around the wellbore on pads that are pressed against the borehole wall. The evolutionary trend from dipmeters to borehole images has been from a few electrodes to a complex array of electrodes on multiple pads. See Hurley, N. F., 2004, Borehole Images, in Asquith, G. and Krygowski, D.: and see Basic Well Log Analysis, 2nd Edition, AAPG Methods in Exploration Series No. 16, p. 151-164. Tools are first run into the hole with the pads closed. At the start of the log run, either four, six, or eight pads are pressed against the borehole wall. The number of pads depends on the logging device. Electrical current is forced into the rock through the electrodes, and sensors measure the current after it interacts with the formation. Raw data include multiple electrode readings, caliper readings from individual pads or pairs of pads, and x-, y-, and z-axis accelerometer and magnetometer readings. Borehole deviation and pad 1 (tool) orientation are determined from the magnetometers. The sample rate for electrode and accelerometer data is very high, normally 120 samples/ft (400 samples/m).
Areal coverage of the borehole face is a function of width of the electrode arrays, number of pads, and borehole diameter. In general, 40 to 80% of the borehole face is imaged in typical boreholes. Non-imaged parts of the borehole appear as blank strips between the pads.
Borehole images are created by assigning color maps to different bins or ranges of resistivity values. Colored pixels are then arranged in their proper geometric position around the wellbore. By convention, low-resistivity features, such as shales or fluid-filled fractures, are displayed as dark colors. High-resistivity features, such as sandstones and limestones, are displayed as shades of brown, yellow, and white.
Two main types of processed borehole images are available: static and dynamic. Static images are those which have had one contrast setting applied to the entire well. They provide useful views of relative changes in rock resistivity throughout the borehole. Dynamic images, which have had variable contrast applied in a moving window, provide enhanced views of features such as vugs, fractures, and bed boundaries. Dynamic images bring out subtle features in rocks that have very low resistivities, such as shales, and very high resistivities, such as carbonates and crystalline rocks.
Another example of electrical borehole Images may run in Oil-Based (non-conductive) Mud, in particular high mud resistivities (greater than 50 ohm-m), typical of oil-based muds, are unsuitable for most electrical borehole images. Since 2001, Schlumberger's OBMI (Oil-Base MicroImager), has been available for oil-based muds. This tool generates borehole images by passing electrical current into the formation from two large electrodes on each pad, which is at a high voltage (about 300V). There is a series of closely spaced buttons, located in two rows of 5 on each of the 4 pads. Borehole images are generated from the potential difference (voltage drop) between the closely spaced electrodes. Wide gaps, corresponding to non-imaged parts of the borehole, are common between pads. Another aspect of Borehole images can be acquired during drilling, e.g., logging-while-drilling or hereafter referred to as “LWD”. Examples of Schlumberger logs are the GeoVision Resistivity (GVR) and Azimuthal Density Neutron (ADN) tools. The GVR uses rotating electrodes, and works in water-based mud. The ADN generates images from azimuthal density readings, and works in any mud. When the tool rotates during drilling, borehole coverage is complete, with no gaps.
Another aspect of Borehole images can be Acoustic borehole images, also known as borehole televiewers, which are based on technology first developed in the 1960's. Zemanek, J., Glenn, E. E., Norton, L. J., and Caldwell, R. L., 1970, Formation evaluation by inspection with the borehole televiewer: Geophysics, v. 35, p. 254-269.
The Ultrasonic Borehole Imager (UBI) is Schlumberger's primary acoustic tool for open-hole applications. The UBI tool, which is centralized in the well, has a rotating transducer that emits and records sound waves that bounce off of the borehole wall. Both acoustic amplitude and travel time are recorded and processed into images. Normally, borehole coverage is 100%, with no gaps in the images. However, poor-quality images may result when the tool is decentralized, or the borehole wall is irregular.
Petrophysical Facies may be considered, among other things, as characteristic signatures on borehole-image logs, such as vugs, and resistive and conductive patches. A particular view by Dehghani et al. in 1999 suggested that zones of enhanced porosity and permeability exist in the vicinity of vugs. Dehghani, K., Harris, P. M., Edwards, K. A., and Dees, W. T., 1999, Modeling a vuggy carbonate reservoir: AAPG Bulletin, v. 83, p. 19-42.
Dehghani et al. (1999) used thin sections, SEM images, and mini-permeability measurements to confirm their concept. Schindler (2005) and Tanprasat (2005) used image analysis of fluorescent-inked core photos to show that swarms of small vugs preferentially exist in the vicinity of large vugs. See Schindler, J., 2005, Quantification of vuggy porosity, Indian Basin field, New Mexico: Unpublished M.S. thesis, Colorado School of Mines, Golden, Colo.; and Tanprasat, S., 2005, Petrophysical analysis of vuggy porosity in the Shu'aiba Formation of the United Arab Emirates: Unpublished M.S. thesis, Colorado School of Mines, Golden, Colo. Such small vugs are below the resolution of the borehole-imaging tool, so they appear as dark regions, rather than as discrete vugs in the image logs. If this is the general case for vuggy carbonates, electrical and acoustic borehole images should have high-conductivity or low-amplitude (dark) zones or halos in the vicinity of vugs. In fact, this feature is commonly observed, for example, as shown in FIG. 3. High-conductivity zones surrounding vugs and enhanced small-scale porosity, known as conductive patches, form the basis for part of Schlumberger's BorTex software, cited in Russell et al. (2002) and Hassall et al. (2004). See Russell, S. D., Akbar, M., Vissapragada, B., and Walkden, G. M., 2002, Rock types and permeability prediction from dipmeter and image logs: Shuaiba reservoir (Aptian), Abu Dhabi: AAPG Bulletin, v. 86, p. 1709-1732; and see Hassall, J. K., Ferraris, P., Al-Raisi, M., Hurley, N. F., Boyd, A., and Allen, D. F., 2004, Comparison of permeability predictors from NMR, formation image and other logs in a carbonate reservoir: SPE preprint 88683, presented at the 11th Abu Dhabi International Petroleum Exhibition and Conference, Abu Dhabi, U.A.E., 10-13 October.
Delhomme (1992) demonstrated the importance of mapping electrically resistive and non-resistive patches in borehole images. Delhomme, J. P., 1992, A quantitative characterization of formation heterogeneities based on borehole image analysis: Trans. 33rd Symposium SPWLA, Paper T. However, his approach worked poorly because of gaps between the pads. He was unable to draw closed contours around regions of high or low resistivity because of uncertainty about the shapes. Fullbore images (FIG. 4) do allow us to draw closed contours around resistive and/or non-resistive regions in borehole images. Such regions provide important measures of heterogeneity, especially in carbonate reservoirs. These regions are generally much larger than the digital rock or core samples we generate from CTscans of rocks, for example. Because of this, we need borehole images to identify if we want to capture decimeter to meter-scale heterogeneities in our flow models.
Regions with characteristic signatures on borehole-image logs, such as vugs, and resistive and conductive patches are herein termed petrophysical facies. Other authors, such as Leduc et al. (2002) and Mathis et al. (2003) call such textural regions electrofacies. See Leduc, J. P., Delhaye-Prat, V., Zaugg, P., and see Mathis, B., 2002, FMI* based sedimentary facies modelling, Surmont Lease (Athabasca, Canada) (abs.): CSPG Annual Convention, Calgary, Alberta, Canada, 10 p.; and see Mathis, B., Leduc, J. P., and Vandenabeele, T., 2003, From the geologists' eyes to synthetic core descriptions: Geological log modeling using well-log data (abs.): AAPG Annual Meeting, Salt Lake City, Utah, 7 p.
Textures represented by the different colors, for example, black, brown, and white (FIG. 4), could be used to define petrophysical facies. Such facies have complex 3D shapes. Conductive patches, if they are zones of enhanced porosity and permeability, and provide regions of flow continuity between vugs.
The published literature has many examples of numerical rocks built using techniques (or digital rock models of rocks and pores) that include reconstructions made from 2D thin sections or scanning-electron microscope (SEM) images, electrofacies interpreted from logs, computer-generated sphere packs, laser-scanning confocal microscopy, and various types of CTscans (conventional, microCT, and synchrotron-computed microtomography).
Bakke and Oren (1997), Oren et al. (1998), and Oren and Bakke (2002) developed a technique that constructs 3D pore networks from 2D thin sections. Numerical Rocks, (http://www.numericalrocks.com/) computes 3D pore models from 2D thin sections. See Bakke, S., and Oren, P.-E., 1997, 3-D pore-scale modeling of sandstones and flow simulations in the pore networks: SPE preprint 35,479, European 3-D Reservoir Modeling Conference, Stavanger, Norway, April 16-17, p. 136-149; Oren, P.-E., Bakke, S., and Arntzen, O. J., 1998, Extending predictive capabilities to network models: SPE Journal, v. 3, p. 324; and Oren, P.-E., and Bakke, S., 2002, Process based reconstruction of sandstones and prediction of transport properties: Transport in Porous Media, v. 46, p. 311-343. This company also uses pore models built from micro-CTscans. Bakke et al. (2002) successfully applied this technique to sucrosic dolomites. Articles by Duey (2008) and Suicmez and Touati (2008) summarize the results of various sandstone pore networks processed by Numerical Rocks. See Duey, R., 2008, Quick analysis answers Heidrun question: Hart Energy Publishing, LP, accessed online at http://www.eandp.info/index2.php?area=article&articleId=767, Mar. 27, 2008; and Suicmez, V. S., and Touati, M., 2008, Pore network modeling: A new technology for SCAL predictions and interpretations: Saudi Arabia Oil and Gas, Issue 5, p. 64-70. Wu et al. (2006) presented a method to generate 3D numerical rock models from 2D thin sections using a third-order Markov mesh. See Wu, K., Van Dijke, M. I. J., Couples, G. D., Jiang, Z., Ma, J., Sorbie, K. S., Crawford, J., Young, I., and Zhang, X., 2006, 3D stochastic modelling of heterogeneous porous media—Applications to reservoir rocks: Transport in Porous Media, v. 65, p. 443-467. Awwiller (2007) developed a technique that simulates more complex sandstones than those described by Oren and Bakke (2002). Awwiller's (2007) patent application, US 2007/0203677 A1 (below), relates to this work. Okabe and Blunt (2004, 2005) generated 3D images from 2D thin sections using multi-point statistics. See Okabe, H., and Blunt, M. J., 2004, Prediction of permeability for porous media reconstructed using multiple-point statistics: Physical Review E, v. 70, p. 066135-1-10; and see Okabe, H., and Blunt, M. J., 2005, Pore space reconstruction using multiple-point statistics: Journal of Petroleum Science and Engineering, v. 46, p. 121-137. Tomutsa and Radmilovic (2003) used ion beam thinning to create multiple 2D serial sections that they used to build 3D models of sub-micron-scale pores. See Tomutsa, L., and Radmilovic, V., 2003, Focused ion beam assisted three-dimensional rock imaging at submicron scale: International Symposium of the Soc. of Core Analysts, Pau, France, September 21-24, Paper SCA2003-47.
Dvorkin et al. (2003) described Digital Rock Physics technology, which consists of pore-scale numerical simulations derived from: (a) 2D thin sections and statistical indicator simulation, or (b) CTscans. See Dvorkin, J., Kameda, A., Nur, A., Mese, A., and Tutuncu, A. N., 2003, Real time monitoring of permeability, elastic moduli and strength in sands and shales using Digital Rock Physics: SPE preprint 82246, presented at the SPE European Formation Damage Conference, The Hague, Netherlands, May 13-14, 7 p. They built 3D models of virtual rock, and did flow simulations using the lattice-Boltzmann method. U.S. Pat. No. 6,516,080 (below) is related to this work.
Leduc et al. (2002) and Mathis et al. (2003) (both noted above) generated “synthetic cores” from a limited number of described cores, conventional openhole logs, and borehole-image logs. Cluster analysis is used on the conventional openhole logs. “Electrofacies,” which are log-based depth intervals of similar lithology, are defined using conventional openhole logs and textural analysis of borehole images. Virtual cores are computed in non-cored wells using “contingency tables.” U.S. Pat. No. 6,011,557 (below) is related to this work.
Vahrenkamp et al. (2008) described mini-models, i.e., reservoir models that are less than 1.0 m3 in size and provide pseudo-properties for volume cells in reservoir-scale models. See Vahrenkamp, V. C., Creusen, A., Tull, S., Farmer, A., Mookerjee, A. and Al Bahry, A., 2008, Multi-scale heterogeneity modelling in a giant carbonate field, northern Oman (abs.): GeoArabia, v. 13, No. 1, p. 248. Mini-models are populated using “principle rock types” (PRT), which “cover and categorize the full range of pore types, sizes, pore-throat size distributions, capillary entry pressures, relative permeabilities, etc.” PRT's are organized into “rock type associations” (RTA), which are based on “sedimentary fabric” determined from borehole-image logs. RTA's are distributed in the reservoir using borehole-image logs, and observed layering, facies models, and seismic data.
Bryant et al. (1993) and Behseresht et al. (2007) described digital rock models that are computer-generated dense random periodic packings of spheres. See Bryant, S., Mellor, D., and Cade, C., 1993, Physically representative network models of transport in porous media: American Institute of Chemical Engineers Journal, v. 39, No. 3, p. 387-396; and see Behseresht, J., Bryant, S. L., and Sepehrnoori, K., 2007, Infinite-acting physically representative networks for capillarity-controlled displacements: SPE preprint 110581, presented at the SPE Annual Technical Conference and Exhibition, Anaheim, Calif., November 11-14, 15 p. Other workers, such as Bosl et al. (1998) and Holt (2001) have generated similar digital rock models for flow experiments. See Bosl, W. J, Dvorkin, J., and Nur, A., 1998, A study of porosity and permeability using a lattice Boltzmann simulation: Geophysical Research Letters, v. 25, p. 1475-1478; and see Holt, R. M., 2001, Particle vs. laboratory modelling in in situ compaction: Physics and Chemistry of the Earth, Part A: Solid Earth and Geodesy, v. 26, Issue 1-2, p. 89-93.
Fredrich et al. (1995) and Fredrich (1999) created 3D images of rocks using laser scanning confocal microscopy. See Fredrich, J. T., Menendez, B., and Wong, T. F., 1995, Imaging the pore structure of geomaterials: Science, v. 268, p. 276-279; and see Fredrich, J. T., 1999, 3D imaging of porous media using laser scanning confocal microscopy with application to microscale transport processes: Physics and Chemistry of the Earth, Part A: Solid Earth and Geodesy, v. 24, Issue 7, p. 551-561. O'Connor and Fredrich (1999) did flow experiments on these numerical rocks using lattice-Boltzmann methods. See O'Connor, R. M., and Fredrich, J. T., 1999, Microscale flow modeling in geologic materials: Physics and Chemistry of the Earth, Part A: Solid Earth and Geodesy, v. 24, Issue 7, p. 611-616.
The most common way to generate pore networks uses various types of CTscans. Vinegar (1986), Wellington and Vinegar (1987), and Withjack et al. (2003) summarized the technology and discussed various applications of X-ray computed tomography. See Vinegar, H. J., 1986, X-ray CT and NMR imaging of rocks: JPT, p. 257-259; see Wellington, S. L., and Vinegar, H. J., 1987, X-ray computerized tomography: JPT, p. 885-898; and see Withjack, E. M., Devier, C., and Michael, G., 2003, The role of X-ray computed tomography in core analysis: SPE preprint 83467, presented at the Western Region/AAPG Pacific Section Joint Meeting, Long Beach, Calif., May 19-24, 2003, 12 p. Siddiqui and Khamees (2005) and Siddiqui et al. (2005) emphasized the use of 3D images of cores and cuttings from conventional and microCTscans. See Siddiqui, S., and Khamees, A. A., 2005, Data visualization challenges for displaying laboratory core and flow data in three-dimensions: SPE preprint 106334, presented at the SPE Technical Symposium of Saudi Arabia, May 14-16, 9 p.; and see Siddiqui, S., and Khamees, A. A., 2005, Data visualization challenges for displaying laboratory core and flow data in three-dimensions: SPE preprint 106334, presented at the SPE Technical Symposium of Saudi Arabia, May 14-16, 9 p. Coles et al. (1996), Fredrich et al. (2006), and Fredrich et al. (2007) used synchrotron-computed microtomography to build numerical 3D models of pore networks in natural and synthetic sandstones. See Coles, M. E., Hazlett, R. D., Muegge, R. L., Jones, K. W., Andrews, B. Dowd, B. Siddons, P., Peskin, A., Spanne, P., and Soll, W. E., 1996, Developments in synchrotron X-ray microtomography with applications to flow in porous media: SPE preprint 36531, presented at the SPE Annual Technical Conference and Exhibition, Denver, Colo., p. 413-424; see Fredrich, J. T., DiGiovanni, A. A., and Noble, D. R., 2006, Predicting macroscopic transport properties using microscopic image data: Journal of Geophysical Research B: Solid Earth, v. 111, Issue 3; and see Fredrich, J. T., Haney, M. M., and White, J. A., 2007, Predicting petrophysical properties using 3D image data (abs.): AAPG Annual Convention, downloaded at http://www.aapg.org. They used lattice-Boltzmann methods to model permeability.
Multi-point (or multiple-point) statistical methods (MPS) are a new family of spatial statistical interpolation algorithms proposed in the 1990's that are used to generate conditional simulations of discrete variable fields, such as geological facies. See Guardiano, F. and Srivastava, R. M., 1993, Multivariate geostatistics: beyond bivariate moments: Geostatistics-Troia, A. Soares. Dordrecht, Netherlands, Kluwer Academic Publications, v. 1, p. 133-144. A training image is a numerical prior geological model that contains the facies structures and relationships believed to exist in realistic reservoirs. Training images are conceptual in nature and can be as simple as a hand-drawn map, or they can be created by computer tools. The original MPS algorithm proposed by Guardiano and Srivastava (1993) built a multiple-point conditional probability distribution function (CPDF) by scanning the training image anew for each simulation node. Because of computer central processing unit (CPU) limitations, this time-consuming algorithm was not practical at that time.
Strebelle (2002) introduced the concept of a search tree, which stores all replicates of patterns found within a template over the training image. See Strebelle, S., 2002, Conditional simulation of complex geological structures using multiple point statistics. Mathematical Geology, v. 34, p. 1-22. Strebelle's (2002) seminal MPS algorithm, called SNESIM, has been used in many applications for reservoir modeling, and has become the reference tool for modeling fluvial channel deposits when combined with rotation and affinity transformations (Zhang, 2002; Caers and Zhang, 2004; Strebelle and Zhang, 2004). See Zhang, T., 2002, Multiple-point simulation of multiple reservoir facies: Unpublished M.S. thesis, Stanford University, California, 163 p.; see Caers, J. and Zhang, T., 2004, Multiple-point geostatistics: A quantitative vehicle for integration of geologic analogs into multiple reservoir models, in M. Grammer, P. M. Harris and G. P. Eberli, eds.: Integration of Outcrop and Modern Analogs in Reservoir Modeling, AAPG. Memoir 80, p. 383-394.; and see Strebelle, S. and Zhang, T., 2004, Non-stationary multiple-point geostatistical models, in Leuangthong, O. and Deutsch, C. V., eds.: Geostatistics, v. 1, p. 235-244.
The seminal MPS algorithm is orders of magnitude faster than Guardiano and Srivastava's (1993) original algorithm, but it is computer random-access memory (RAM) demanding, especially in 3D for a large training image. See Guardiano, F. and Srivastava, R. M., 1993, Multivariate geostatistics: beyond bivariate moments: Geostatistics-Troia, A. Soares. Dordrecht, Netherlands, Kluwer Academic Publications, v. 1, p. 133-144. This RAM limitation in 3D requires compromises that may lead to inadequate shape reproduction of 3D objects. The RAM limitation also prevents us from considering too many categories or classes jointly, thus limiting seminal MPS algorithm to the simulation of categorical variables.
In order to deal with both categorical and continuous variable training images and reduce RAM cost and improve shape reproduction in 3D applications, an MPS algorithm such as FILTERSIM (Zhang 2006a). See Zhang, T., 2006a, Filter-based training image pattern classification for spatial pattern simulation: Unpublished Ph.D. dissertation, Stanford University, California, 153 p. The FILTERSIM algorithm applies a set of local filters to the training image, which can be either categorical or continuous, to group local patterns into pattern classes. It then proceeds to simulate patterns on the basis of that classification. A filter is a local template (window) with a set of weights associated to each pixel location of the template. Applying a filter to a local pattern results in a filter score, the score is viewed as a numerical summary of that local pattern. A set of default or use-defined filters is designed such that each filter can record different aspects of the training pattern seen within the template. These filters are used to transform training patterns into a filter score space. This pattern scoring provides a dimension reduction of patterns. By partitioning that score space of limited dimension, similar training patterns are classified based on their filter scores.
The seminal MPS algorithm is orders of magnitude faster than Guardiano and Srivastava's (1993) original algorithm, but it is computer random-access memory (RAM) demanding, especially in 3D for a large training image. This RAM limitation in 3D requires compromises that may lead to inadequate shape reproduction of 3D objects. The RAM limitation also prevents us from considering too many categories or classes jointly, thus limiting seminal MPS algorithm to the simulation of categorical variables. The seminal MPS algorithm searches for exact replicates of the conditioning data event, builds the reservoir model one pixel at a time, conditioned to a multiple-point data event, and does not allow any filtering or averaging of the patterns found in the training image.
In order to deal with both categorical and continuous variable training images and reduce RAM cost and improve shape reproduction in 3D applications, a new MPS algorithm named FILTERSIM was proposed by Zhang (2006a). The FILTERSIM algorithm applies a set of local filters to the training image, which can be either categorical or continuous, to group local patterns into pattern classes. It then proceeds to simulate patterns on the basis of that classification. A filter is a local template (window) with a set of weights associated to each pixel location of the template. Applying a filter to a local pattern results in a filter score, the score is viewed as a numerical summary of that local pattern. A set of default or use-defined filters is designed such that each filter can record different aspects of the training pattern seen within the template. These filters are used to transform training patterns into a filter score space. This pattern scoring provides a dimension reduction of patterns. By partitioning that score space of limited dimension, similar training patterns are classified based on their filter scores.
The FILTERSIM algorithm starts with a classification of local training patterns in a filter score space of reduced dimension. Simulation proceeds along a sequential path through the simulation space, by determining which pattern class is most similar to the local conditioning data event, sampling a specific pattern from the pattern class, and then patching the sampled pattern onto the image at the simulation sites. The simulation random path and the sampling of patterns from pattern classes allow for different simulated realizations, yet all are conditional to the same original data. Because of the dimension reduction brought by the filter summaries of any pattern, and because patterns are grouped into classes, the algorithm is fast and reasonable in terms of RAM demand.
The seminal MPS algorithm and FILTERSIM algorithm are able to honor absolute or so-called “hard” constraints from data acquired in wells or outcrops, and conditional or “soft” constraints from seismic data, facies probability fields, and rotation and affinity (or scale) constraint grids. All of these data are used in the stochastic modeling process to generate 1D, 2D, or 3D maps of geological facies or rock properties. Because there is a random component involved in MPS simulations, individual realizations of property fields created by MPS algorithms differ, but the ensemble of realizations provides geoscientists and reservoir engineers with improved quantitative estimates of the spatial distribution and uncertainty of geological facies in a modeled reservoir volume. Moreover, these algorithms honor both hard and soft input data constraints (Zhang, 2006a). See Zhang, T., Switzer P., and Journel A., 2006b, Filter-based classification of training image patterns for spatial pattern simulation: Mathematical Geology, v. 38, p. 63-80.
Six directional 2D default filters that are typically used in FILTERSIM (Zhang, 2006a; Zhang et al., 2006b) (both noted above). There are three types of filters: average filter, gradient filter and curvature filter, and each type of filter are used for both horizontal and vertical directions. Average filters aim at localizing features; gradient filters are used to detect feature boundaries by highlighting the contrast of different features (the first-order difference); curvature filters supply the second-order difference of features.
In order to reflect large-scale structure, multi-grid simulation is used. This progressively simulates each level of the multi-grid from coarser to finer with the finer-grid simulation being constrained by previously simulated values at coarser grids. At each level of the simulation, rescaled filters are applied over the respective grid (Zhang, 2006a).
There are two types of training images: one with a very limited number of categories and another for continuous variables such as reservoir petrophysical properties. Multipoint geostatistical methods require 1D, 2D, or 3D grids of training images as prior conceptual geological models that contain patterns of the spatial attributes under study. The shapes of different features appearing on the images are supposed to represent a model of real geological features, with each category typically representing a different geological facies or different kind of geological body. Training images are typically required to contain “stationary” patterns, i.e., the patterns must be independent of their location in space (invariant according to any translation) and must be repetitive over the training image area. In the case of training images used for geological modeling, this stationarity can consist, but is not limited to, geological object orientation stationarity (where directional objects/features do not rotate across the image) and geological scale stationarity (where the size of objects/features on the image does not change across the image) (Caers and Zhang, 2004). See Caers, J. and Zhang, T., 2004, Multiple-point geostatistics: A quantitative vehicle for integration of geologic analogs into multiple reservoir models, in M. Grammer, P. M. Harris and G. P. Eberli, eds.: Integration of Outcrop and Modern Analogs in Reservoir Modeling, AAPG. Memoir 80, p. 383-394.
An issue raised implicitly by current MPS algorithms is how to generate training images. Training images are supposed to model or reproduce real geological features and should as much as possible be derived from existing geologically meaningful images. This requires research on statistical and image-processing methods that will allow use of images from any source, e.g., hand-drawn sketches, aerial photographs, satellite images, seismic volumes, geological object models, physical scale models, or forward geological process models. Compared to the creation of continuously variable training images, generating categorically variable training images is easier. An object-based approach is commonly used to generate training images with categorical variables. A region-based approach, combined with adding desired constraints, can be used to generate continuously variable training mages (Zhang et al., 2005). See Zhang, L., Nair, N., Jennings, J. W., and Bryant, S. L., 2005, Models and methods for determining transport properties of touching-vug carbonates: SPE preprint 96027, presented at the SPE Annual Technical Conference and Exhibition, Dallas, Tex., October 9-12, 9 p.
In order to perform MPS simulation, training images must be stationary. However, in most reservoir modeling applications, geological sediments show non-stationary patterns/features, which reflect reservoir heterogeneities and anisotropies of sedimentation.